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Linear rigidity of stationary stochastic processes
Alexander I. Bufetov, Yoann Dabrowski, Yanqi Qiu
To cite this version:
arXiv:1507.00670v1 [math.PR] 2 Jul 2015
Linear rigidity of stationary stochastic
processes
Alexander I. Bufetov
∗Yoann Dabrowski
†Yanqi Qiu
‡Abstract
We consider stationary stochastic processes Xn, n ∈ Z such that X0
lies in the closed linear span of Xn, n6= 0; following Ghosh and Peres, we
call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be rigid, that the spectral density vanish at zero and belong to the Zygmund classΛ∗(1). We next give
sufficient condition for stationary determinantal point processes on Z and on R to be rigid. Finally, we show that the determinantal point process on R2 induced by a tensor square of Dyson sine-kernels is not linearly rigid.
1
Introduction
This paper is devoted to rigidity of stationary determinantal point processes. Recall that stationary determinantal point processes are strongly chaotic: they have the Kolmogorov property (Lyons [9]) and the Bernoulli property (Lyons and Steif [10]); and they satisfy the Central Limit Theorem (Costin and Lebowitz [2], Soshnikov[13]). On the other hand, Ghosh [5] and Ghosh-Peres [6] proved, for the determinantal point processes such as Dyson sine process and Ginibre point pro-cess, that number of particles in a finite window is measurable with respect to the
∗bufetov@mi.ras.ru; Aix-Marseille Universit´e, Centrale Marseille, CNRS, I2M, UMR7373, 39
Rue F. Joliot Curie 13453, Marseille Cedex; Steklov Institute of Mathematics, Moscow; Institute for Information Transmission Problems, Moscow; National Research University Higher School of Economics, Moscow.
†dabrowski@math.univ-lyon1.fr, Universit´e de Lyon, Universit´e Lyon 1, Institut Camille
Jor-dan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France.
‡yqi.qiu@gmail.com, Aix-Marseille Universit´e, Centrale Marseille, CNRS, I2M, UMR7373,
completion of the sigma-algebra describing the configurations outside that finite window. Their argument is spectral: they construct, for any smallε, a compactly
supported smooth functionϕε, such thatϕεequals1 in a fixed finite window and
the linear statistic corresponding toϕεhas variance smaller thanε.
In the same spirit, we consider general stationary stochastic processes (in broad sense)Xn, n ∈ Z such that X0 lies in the closed linear span ofXn,n 6= 0;
following Ghosh and Peres, we call such processes linearly rigid. Using a crite-rion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be rigid, that the spectral density vanish at zero and belong to the Zygmund classΛ∗(1). We next give sufficient condition for stationary determinantal point
processes on Z and on R to be rigid. Finally, we show that the determinantal point process on R2 induced by a tensor square of Dyson sine-kernels is not linearly rigid.
We now turn to more precise statements. LetX = {Xn : n ∈ Zd} be a
multi-dimensional time stationary stochastic process of real-valued random variables defined on a probability space (Ω, P). Let H(X) ⊂ L2(Ω, P) denote the closed
subspace linearly spanned by {Xn : n ∈ Zd} and let ˇH0(X) denote the one
linearly spanned by{Xn: n ∈ Zd\ {0}}.
Definition 1.1. The stochastic processX is said to be linearly rigid if
X0 ∈ ˇH0(X). (1)
LetConf(Rd) be the set of configurations on Rd. For a bounded Borel subset
B ⊂ Rd, we denoteN
B : Conf(Rd) → N ∪ {0} the function defined by
NB(X) := the cardinality of B ∩ X .
The spaceConf(Rd) is equipped with the Borel algebra which is the smallest
σ-algebra making allNB’s measurable. Recall that a point process with phase space
Rd is, by definition, a Borel probability measure on the spaceConf(Rd). For the
background on point process, the reader is referred to Daley and Vere-Jones’ book [3].
Given a stationary point process on Rdandλ > 0, we introduce the stationary
stochastic processN(λ) = (N(λ)
n )n∈Zd by the formula
Definition 1.2. A stationary point process P on Rdis called linearly rigid, if for anyλ > 0, the stationary stochastic process N(λ) = (N(λ)
n )n∈Zd is linearly rigid,
i.e.,
N0(λ) ∈ ˇH0(N(λ)).
The above definition is motivated by the definition due to Ghosh and Peres of rigidity of point processes on Rd, see [5] and [6]. Given a Borel subsetC ⊂ Rd,
we will denote
FC = σ({NB : B ⊂ C, B bounded Borel})
the σ-algebra generated by all random variables of the form NB whereB ⊂ C
ranges over all bounded Borel subsets ofC. Let P be a point process on R, i.e., P
is a Borel probability onConf(Rd), and denote FP
C for the P-completion of FC.
Definition 1.3 (Ghosh [5], Ghosh-Peres [6]). A point process P on Rd is called
rigid, if for any bounded Borel set B ⊂ Rdwith Lebesgue-negligible boundary
∂B, the random variable NB is FPRd\B-measurable.
Remark 1.1. Of course, in the above definition, it suffices to take Borel setsB of
the form[−γ, γ)dforγ > 0, cf. [6].
A linear rigid stationary point process on Rd is of course rigid in the sense of Ghosh and Peres. Observe that proofs for rigidity in [5], [6] and [1] in fact establish linear rigidity. We would like also to mention a notion of insertion-deletion tolerance studied by Holroyd and Soo in [7], which is in contrast to the notion of rigidity property.
2
The Kolmogorov criterion for linear rigidity
In this note, the Fourier transform of a functionf : Rd→ C is defined as
b f (ξ) =
Z
Rd
f (x)e−i2πx·ξdx.
Denote by Td = Rd/Zdthed-dimensional torus. In what follows, we identify Td
with [−1/2, 1/2)d. The Fourier coefficients of a measure µ on Td are given, for
anyk ∈ Zd, by the formula
ˆ µ(k) =
Z
Td
Denote byµX the spectral measure ofX, i.e.,
∀k ∈ Zd, E(X0Xk) = E(XnXn+k) = Z
Td
e−i2πk·θdµX(θ) = ˆµX(k). (3)
Recall that we have the following natural isometric isomorphism
H(X) ≃ L2(Td, µX), (4)
by assigning toXn∈ H(X) the function θ 7→ ei2πn·θ∈ L2(Td, µX).
LetµX = µa+ µsbe the Lebesgue decomposition ofµX with respect to the
normalized Lebesgue measure m(dθ) = dθ1· · · dθd on Td, i.e., µa is absolutely
continuous with respect tom and µsis singular tom. Set
ωX(θ) :=
dµa
dm(θ). Lemma 2.1 (The Kolmogorov Criterion). We have
dist(X0, ˇH0(X)) = Z Td ω−1X dm −1/2 . The right side is to be interpreted as zero ifRTdω
−1
X dm = ∞.
When the measureµ is assumed to be absolutely continuous with respect to m, Lemma2.1is a result of Kolmogorov, see Remark 5.17 in Lyons-Steif [10].
Corollary 2.2. The stationary stochastic processX = (Xn)n∈Zd is linearly rigid
if and only if Z
Td
ωX−1dm = ∞.
Proof of Lemma2.1. We follow the argument of Lyons-Steif [10]. By the Lebesgue decomposition of µ, we may take a subset A ⊂ Td of full Lebesgue measure
m(A) = 1, such that µa(A) = 1 and µs(A) = 0.
Denote
L0 = spanL
2(Td,µ
X)[ei2πn·θ: n 6= 0].
By the isometric isomorphism (4), it suffices to show that
where1 is the constant function taking value 1. Write 1 = p + h, such that p ⊥ L0, h ∈ L0.
Modifying, if necessary, the values ofp and h on a µ-negligible subset, we may
assume that 1 = p(θ) + h(θ) for all θ ∈ Td. Sincep ⊥ L0, we have 0 = hp, ei2πn·θi L2(dµ)= Z Td
p(θ)e−i2πn·θdµ(θ), for any n ∈ Zd\ 0. (6)
Therefore, the complex measure p · dµ is a multiple of Lebesgue measure, i.e.,
there existsξ ∈ C, such that
p · dµ = ξdm.
It follows thatp must vanish almost everywhere with respect to the singular
com-ponentµsofµ, and p(θ)ωX(θ) = ξ for m-almost every θ ∈ Td. Thus we have
kpkL2(dµ) = kpkL2(dµa), (7)
and
h(θ) = 1 − ξωX(θ)−1 form-almost every θ ∈ Td. (8)
Case 1: RTdω −1 X dm < ∞. Define a functionf : Td → C by f = ω−1 X χA. Thenf ∈ L2(dµ) ⊖L0. Indeed, kf k2L2(dµ) = Z Td ω−2X χAdµ = Z Td ωX−2dµa = Z Td ωX−1dm < ∞.
And, for alln ∈ Zd\ 0,
hf, ei2πn·θiL2(dµ) = Z Td ωX(θ)−1χA(θ)e−i2πn·θdµ(θ) = Z Td e−i2πn·θdm(θ) = 0.
It follows thatf ⊥ h, i.e.,
By (8), we get Z Td (1 − ξω−1X )dm = 0, and hence ξ = ( Z Td ωX−1dm)−1. It follows that dist(1, L0)2 = kpk2L2(dµ) = kpk2L2(dµ a) = ξ 2 Z Td ωX−2ωXdm = ξ.
This shows the desired equality (5).
Case 2: RTdω
−1
X dm = ∞.
We claim thatξ = 0. If the claim were verified, then we would get the desired
identity in this case
dist(1, L0) = 0.
So let us turn to the proof of the claim. We argue by contradiction. Ifξ 6= 0, then p 6= 0 and
kpk2L2(dµ)= kpk2L2(dµa)= ξ2kωX−1k2L2(dµa)= ξ2 Z
Td
ωX−1dm = ∞.
This contradicts the fact thatp ∈ L2(dµ).
Remark 2.1. The same argument shows that, in the case of one-dimensional time,
the following assertions are equivalent:
• Pnk=−nXk∈span{Xj : |j| ≥ n + 1};
• for any α1, · · · , αn∈ (−1/2, 1/2) \ {0}, we have
Z
T
Qn
j=1|ei2πθ− ei2παj|2|ei2πθ − e−i2παj|2
ωX(θ)
dm(θ) = ∞.
Denote byCov(U, V ) the covariance between two random variables U and V : Cov(U, V ) = E(UV ) − E(U)E(V ).
IfX = (Xn)n∈Zd is a stochastic process such that X
n∈Zd
|Cov(X0, Xn)| < ∞, (9)
then we may define a continuous function on Tdby the formula
ωX(θ) :=
X
n∈Zd
Cov(X0, Xn)ei2πn·θ. (10)
Lemma 2.3. Let X = (Xn)n∈Zd be a stationary stochastic process satisfying
condition (9). Then we have the following explicit Lebesgue decomposition of
µX:
µX = (EX0)2· δ0+ ωX · m, (11)
where δ0 is the Dirac measure on the point0 ∈ TdandωX is the function on Td
defined by (10).
Proof. Note that, under the assumption (9), the functionωX(θ) is well-defined and
continuous on Td. For proving the decomposition (11), it suffices to show that the Fourier coefficients ofµX coincide with those ofνX := (EX0)2· δ0+ ωX· m. But
ifn ∈ Zd, then
ˆ
νX(n) = (EX0)2+ Cov(X0, Xn) = E(X0Xn) = ˆµX(n).
The lemma is completely proved.
3
A sufficient condition for linear rigidity
Theorem 3.1. LetX = (Xn)n∈Z be a stationary stochastic process. If
sup N ≥1 N X |n|≥N |Cov(X0, Xn)| < ∞, (12) and X n∈Z Cov(X0, Xn) = 0. (13)
Remark 3.1. The condition (12) is a sufficient condition such that the spectral density ωX is a function in the Zygmund class Λ∗(1), see below for definition.
The condition (13) implies in particular thatωX vanishes at the point0 ∈ T.
We shall apply a result of F. M´oricz [12, Thm. 3] on absolutely convergent Fourier series and Zygmund class functions. Recall that a continuous 1-periodic
functionϕ defined on R is said to be in the Zygmund class Λ∗(1), if there exists a
constantC such that
|ϕ(x + h) − 2ϕ(x) + ϕ(x − h)| ≤ Ch (14) for allx ∈ R and for all h > 0.
Theorem 3.2 (M´oricz, [12]). If{cn}n∈Z ∈ C is such that
sup N ≥1 N X |n|≥N |cn| < ∞, (15)
then the functionϕ(θ) =Pn∈Zcnei2πnθis in the Zygmund classΛ∗(1).
Proof of Theorem3.1. First, in view of (10), our assumption (13) implies
ωX(0) = 0.
Next, by Theorem3.2, under the assumption (12), we have
ωX ∈ Λ∗(1).
Since all Fourier coefficients ofωX are real, we have
ωX(θ) = ωX(−θ).
Consequently, there existsC > 0, such that
ωX(θ) = ωX(θ) + ωX(−θ) 2 = ωX(θ) + ωX(−θ) − 2ωX(0) 2 ≤ C|θ|, whence Z T ωX−1dm = ∞,
and the stochastic process X = (Xn)n∈Z is linearly rigid by the Kolmogorov
4
Applications to stationary determinantal point
pro-cesses
In this section, we first give a sufficient condition for linear rigidity of stationary determinantal point processes on R and then give an example of a very simple stationary, but not linearly rigid, determinantal point process on R2. We briefly recall the main definitions. Let B ⊂ Rd be a bounded Borel subset. Let K
B :
L2(Rd) → L2(Rd) be the operator of convolution with the Fourier transformχc B
of the indicator functionχB. In other words, the kernel ofKBis
KB(x, y) =χcB(x − y). (16)
In particular, ifd = 1 and B = (−1/2, 1/2), then we find the well-known Dyson
sine kernel
Ksine(x, y) =
sin(π(x − y)) π(x − y) .
Note that we always haveKB(x, x) = KB(0, 0).
Denote by PKB the determinantal point process induced byKB. For the
back-ground on the determinantal point processes, the reader is referred to [8], [9], [11], [13].
Proposition 4.1. Let PKB be the stationary determinantal point process on R
d
induced by the kernelKB in (16). For anyλ > 0, denote by N(λ) = (Nn(λ))n∈Zd
the stationary stochastic process associated to PKB as in (2). Then X n∈Zd |Cov(N0(λ), Nn(λ))| < ∞ (17) and X n∈Zd Cov(N0(λ), Nn(λ)) = 0. (18)
Proof. Fix a number λ > 0, for simplifying the notation, let us denote Nn(λ) by
Nn. Denote for anyn ∈ Zd,
Qn= nλ + [−λ/2, λ/2)d.
By definition of a determinantal point process, we have
E(Nn) = E(N0) = Z
Q0
Ifn 6= 0, we have E(N0Nn) = Z Z χQ0(x)χQn(y) KKBB(x, x) K(y, x) KBB(x, y)(y, y) dxdy = λ2dK B(0, 0)2− ZZ Q0×Qn |KB(x, y)|2dxdy, whence Cov(N0, Nn) = − Z Z Q0×Qn |KB(x, y)|2dxdy. (19) We also have E(N02) = E " X x,y∈X χQ0(x)χQ0(y) # = E " X x∈X χQ0(x) # + E " X x,y∈X,x6=y χQ0(x)χQ0(y) # = Z Q0 KB(x, x)dx + ZZ χQ0(x)χQ0(y) KKBB(x, x) K(y, x) KBB(x, y)(y, y) dxdy = λdKB(0, 0) + λ2dKB(0, 0)2− Z Z Q0×Q0 |KB(x, y)|2dxdy, whence Cov(N0, N0) = Var(N0) = λdKB(0, 0) − ZZ Q0×Q0 |KB(x, y)|2dxdy. (20)
Now recall thatKBis an orthogonal projection. Thus we have
KB(0, 0) = KB(x, x) = Z |KB(x, y)|2dy = X n∈Zd Z Qn |KB(x, y)|2dy. (21)
The identities (19), (20) and (21) imply that
X n∈Zd Cov(N0, Nn) = λdKB(0, 0) − Z Q0 dxX n∈Zd Z Qn |KB(x, y)|2dy = λdKB(0, 0) − λdKB(0, 0) = 0.
Remark 4.1. By Lemma 2.3 and Proposition4.1, we see that for any stationary determinantal point process induced by a projection, the spectral density of the associated stochastic processN(λ)always vanishes at0.
4.1
Stationary determinantal point processes on R
Theorem 4.2. Assume thatB ⊂ R satisfies sup R>0 R Z |ξ|≥R |χcB(ξ)|2dξ < ∞. (22)
Then the stationary determinantal point process PKB is linearly rigid.
Proof. By definition of linear rigidity, we need to show that for any λ > 0, the
stochastic processN(λ) = (N(λ)
n )n∈Zis linearly rigid. As in the proof of
Proposi-tion4.1, we denoteNn(λ)byNn. By Theorem3.1, it suffices to show that
sup N ≥1 N X |n|≥N |Cov(N0, Nn)| < ∞, (23) and X n∈Z Cov(N0, Nn) = 0. (24)
By Proposition 4.1, the identity (24) holds in general case. It remains to prove (23). By (19), we have sup N ≥1 N X |n|≥N |Cov(N0, Nn)| = sup N ≥1 N ZZ S |n|≥N Qn |χbB(x − y)|2dxdy ≤ sup N ≥1 λN Z |ξ|≥(N −1)λ |χbB(ξ)|2dξ < ∞
where in the last inequality, we used our assumption (22). Theorem4.2is proved completely.
Remark 4.2. WhenB is a finite union of finite intervals on the real line, the rigidity
4.2
Tensor product of sine kernels
In higher dimension, the situation becomes quite different. Let
S = I × I = (−1/2, 1/2) × (−1/2, 1/2) ⊂ R2.
Then the associate kernelKS has a tensor form: KS = Ksine⊗ Ksine, that is, for
x = (x1, x2) and y = (y1, y2) in R2, we have KS(x, y) = Ksine(x1, y1)Ksine(x2, y2) = sin(π(x1− y1)) π(x1− y1) sin(π(x2− y2)) π(x2− y2) . Proposition 4.3. The determinantal point process PKS is not linearly rigid. More
precisely, let N(1) = (N(1)
n )n∈Z2 be the stationary stochastic process given as in
Definition1.2, then
N0(1) ∈ ˇ/ H0(N(1)).
To prove the above result, we need to introduce some extra notation. First, we define the multiple Zygmund classΛ∗ as follows. A continuous functionϕ(x, y)
periodic in each variable with period 1 is said to be in the multiple Zygmund
classΛ∗(1, 1) if for the double difference difference operator ∆2,2of second order
in each variable, applied to ϕ, there exists a constant C > 0, such that for all x = (x1, x2) ∈ (−1/2, 1/2) × (−1/2, 1/2) and h1, h2 > 0, we have |∆2,2ϕ(x1, x2; h1, h2)| ≤ Ch1h2, (25) where ∆2,2ϕ(x1, x2; h1, h2) := ϕ(x1+ h1, x2+ h2) + ϕ(x1− h1, x2 + h2) + ϕ(x1+ h1, x2− h2) + ϕ(x1− h1, x2− h2) − 2ϕ(x1+ h1, x2) − 2ϕ(x1− h1, x2) − 2ϕ(x1, x2+ h2) − 2ϕ(x1, x2− h2) + 4ϕ(x1, x2).
The following result is due to F¨ul¨op and M´oricz [4, Thm 2.1 and Rem. 2.3]
Theorem 4.4 (F¨ul¨op-M´oricz). If{cjk}j,k∈Z ∈ C is such that
sup N ≥1,M ≥1 MN X |j|≥N,|k|≥M |cjk| < ∞, (26)
then the function
ϕ(θ1, θ2) =
X
j,k∈Z
cjkei2π(jθ1+kθ2)
Let us turn to the study of the density functionωN(1).
Lemma 4.5. There existsc > 0, such that for any θ1, θ2 ∈ (−1/2, 1/2), we have
ωN(1)(θ1, θ2) ≥ c(|θ1| + |θ2|).
Proof. To make notation lighter, in this proof we simply writeω for ωN(1).
DenoteSn = S × (n + S) where n + S := (−1/2 + n1, 1/2 + n1) × (−1/2 +
n2, 1/2 + n2). By the same argument as in the proof of Theorem 4.2, we obtain
that for anyn = (n1, n2) ∈ Z2\ 0,
b ω(n) = − Z Sn |KS(x, y)|2dxdy, and b ω(0) = KS(0, 0) − Z S0 |KS(x, y)|2dxdy.
The following properties can be easily checked.
• Pn∈Z2ω(n) = 0.b
• ω(εb 1n1, ε2n2) =ω(nb 1, n2), where ε1, ε2 ∈ {±1}.
• there exist c, C > 0, such that c (1 + n2 1)(1 + n22) ≤ |ω(nb 1, n2)| ≤ C (1 + n2 1)(1 + n22) .
For instance, Pn∈Z2ω(n) = 0 follows from Propositionb 4.1. These properties
combined with Theorem4.4yield that
• ω(0, 0) = 0.
• ω(ε1θ1, ε2θ2) = ω(θ1, θ2) for any ε1, ε2 ∈ {±1} and θ1, θ2 ∈ (−1/2, 1/2).
• the function ω(θ1, θ2) is in the multiple Zygmund class Λ∗(1, 1).
Hence there existsC > 0, such that
Lemma 4.6. There existsc > 0, such that
ω(θ1, 0) ≥ c|θ1| and ω(0, θ2) ≥ c|θ2|. (28)
Let us postpone the proof of Lemma4.6and proceed to the proof of Lemma
4.5. The inequalities (27) and (28) imply that
ω(θ1, θ2) ≥ c(|θ1| + |θ2|) − C|θ1θ2|.
To prove the lower bound of type as in the lemma, it suffices to prove it when|θ1|
and|θ2| are small enough, for instance, 2C|θ1| ≤ c, then we have
ω(θ1, θ2) ≥
c
2(|θ1| + |θ2|).
Now let us turn to the
Proof of Lemma4.6. By symmetry, it suffices to prove that there exists c > 0,
such that ω(θ1, 0) ≥ |θ1|. To this end, let us denote ω1(θ1) := ω(θ1, 0). Then
ω1(0) = 0 and there exists c > 0 such that if k 6= 0, then
b ω1(k) < 0 and |ωb1(k)| ≥ c/(1 + k2), Indeed, we have ω1(θ1) = X k∈Z X n2∈Z b ω(k, n2)ei2πkθ1,
ifk 6= 0, thenω(k, nb 2) < 0 and hence
|ωb1(k)| = X n2∈Z |ω(k, nb 2)| ≥ X n2∈Z c (1 + n2 2)(1 + k2) ≥ c ′ 1 + k2.
Note also thatω1(0) = ω(0, 0) = 0, hence
X
k∈Z
b
It follows that ω1(θ1) = X k∈Z b ω1(k)ei2πkθ1 = X k∈Z b ω1(k)( ei2πkθ1+ e−i2πkθ1 2 − 1) = X k∈Z,k6=0 −bω1(k)(1 − cos(2πkθ1)) = X k∈Z,k6=0 |ωb1(k)|(1 − cos(2πkθ1)) ≥ c′′ ∞ X j=1 1 (2j − 1)2(1 − cos(2π(2j − 1)θ1)).
Combining with the classical formulae
∞ X j=1 1 (2j − 1)2 = π2 8 , |α| = 1 4 − 2 π2 ∞ X j=1 cos(2(2j − 1)πα) (2j − 1)2 , for α ∈ (−1/2, 1/2); we obtain that ω1(θ1) ≥ c′′ π2 2 |θ1|.
Proof of Proposition4.3. By Lemma2.1, it suffices to show that
Z
T2
ωN−1(1)dm < ∞. (29)
Acknowledgements
The authors would like to thank Guillaume Aubrun for many valuable discussions. A.B. and Y.Q. are supported by A*MIDEX project (No. ANR-11-IDEX-0001-02), financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency (ANR). A. B. is also supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by the ANR under the project “VALET” of the Programme JCJC SIMI 1, and by the RFBR grants 11-01-00654, 12-01-31284, 12-01-33020, 13-01-12449.
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